Homeschooling Corner: Flying Kites

We had a lovely, windy day, so we grabbed the kites, invited the neighbors, and headed out to the park.

Homeschooling does put additional responsibility on the parents to help their kids socialize. That doesn’t mean homeschooled kids are necessarily at a disadvantage viz their typically-schooled peers when it comes to comes to socializing (I went to regular school and still managed to be terribly socialized;) it’s just one more thing homeschooling parents have to keep in mind. So I am glad that we’ve had the good luck recently to make several friends in the neighborhood.

I’ve been looking for good, educational YouTube channels. Now I haven’t watched every video on these channels and I make no guarantees, but they seem good so far:

Welch Labs:

Welch Labs also has a website with a free downloadable workbook that accompanies their videos about imaginary numbers. It’s a good workbook and I’m working through it now.

TedEd, eg:

VSauce, eg:

Numberphile, eg:

The King of Random, eg:

We finished DK’s Coding in Scratch Projects Workbook and started Coding in Scratch: Games Workbook, which is slightly more advanced (longer projects.)

The Usborne Times Tables Activity Book is a rare find: a book that actually makes multiplication vaguely fun. Luckily there’s no one, set age when kids need to learn their multiplication tables–so multiple kids can practice their tables together.

In math we’ve also been working with number lines, concept like infinity (countable and uncountable,) infinitesimals, division, square roots, imaginary numbers, multi-digit addition and subtraction, graphing points and lines on the coordinate plane, and simple functions like Y=X^2. (Any kid who has learned addition, subtraction, multiplication and division can plot simple functions.)

We started work with the cuisenaire rods, which I hope to continue–I can’t find our set on Amazon, but these are similar. We’re also using Alexander Warren’s book You can Count on it: A Mentor’s Arithmetic Patterns for Elementary Students for cusienaire activites.

If you’re looking for board game to play with elementary-aged kids, Bejeweled Blitz is actually pretty good. Two players compete to place tiles on the board to match 3 (or more) gems, in a row or up and down. (A clever play can thus complete two rows at once.) We play with slightly modified rules. (Note: this game is actually pretty hard for people who struggle with rotating objects in their heads.)

Picture Sudoku is fun for little kids (and probably comes in whatever cartoon characters you like,) while KenKen and magic squares and the like are good for older kids (I always loved logic puzzles when I was a kid, so I’d like to get a book of those.)

I’ve found a website called Memrise which seems good for learning foreign languages if you don’t have access to a tutor or know somene who speaks the language you want to learn. They probably have an app for phones or tablets, so kids could practice their foreign langauge on-the-go. (Likewise, I should stow our spelling book in the car and use car rides as a chance to quiz them.)

And of course we’re still reading Professor Astro Cat/working in the workbook, which involves plenty of writing.

For Social Studies we’ve been reading about fall holidays.

Hope you all have a lovely October! What are some of your favorite educational videos?

 

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Homeschooling Corner: The Things we Played

I’m a really boring person who gets excited about finding math workbooks at the secondhand shop. I got lucky this week and snagged two math and 1 science workbooks, plus Bedtime Math 2 at the library. Since new workbooks/manipulatives/materials can be pricey,* I’ve been keeping an eye out for good deals for, well, pretty much my kids’ whole lives. For example, a few years ago I found Hooked on Math ($45 on Amazon) at Goodwill for a couple of bucks; I found some alphabet flashcards at a garage sale for 50c.

I’m also lucky to have several retired teachers in the family, so I’ve “inherited” a nice pile of teaching materials, from tangrams to fractions.

*That said, sometimes you need a particular workbook now, not whenever one shows up at the second hand shop, so thankfully plenty of workbooks are actually pretty cheap.

But full “curriculums” can be pretty expensive–for example, Saxon Math plus manipulatives runs about $200; a Lifepack 4 or 5-subject curriculum is about $320; Montessori math kit: $250; Horizons: $250. I have no idea if these are worth the money or not.

So I’m glad I already have most of what I need (for now.)

This week we started typing (I went with the first website that came up when I searched for “typing tutor” and so far it’s gone well.) We finished Bedtime Math and moved on to Bedtime Math 2. (We’re also working out of some regular old math books, as mentioned above.)

In science we’re still reading Professor Astro Cat’s Frontiers of Space (today we discussed eclipses,) and we started Professor Astro Cat’s Intergalactic Workbook, which has been fun so far. It has activities based on space gloves, weightlessness, Russian phrases (used on the International Space Station,) Morse Code, etc.

(The gloves activity was difficult for youngest child–in retrospect, one pair of glove would have been sufficient. Eventually they got frustrated and started using their feet instead of hands to complete the activities.)

Professor Astro Cat has therefore been the core of our activities this week.

To keep things light, I’ve interspersed some games like Trucky3, Perplexus, and Fraction Formula. They’re also useful when one kid has finished an activity and another hasn’t and I have to keep them occupied for a while.

Coding continues apace: learned about loops this week.

Spelling is one of our weak points, so I want to do at least some spelling each day, (today we spelled planets’ names) but I’m not sure what the best approach is. English spelling is pretty weird.

Homeschooling Corner

Welcome! Highly unscientific polling has revealed an interest in a regular or semi-regular feature focused on homeschooling.

Note that I am NOT some homeschooling guru with years of experience. We are just beginning, so I want some other people to discuss things with. I don’t have a curriculum picked out nor a coherent “philosophy,” but I am SO EXCITED about all of the things I have to teach I couldn’t even list them all.

I was thinking of starting with just a focus on what has been successful this week–which books/websites/projects we liked–and perhaps what was unsuccessful. I invite all of you to come and share your thoughts, ideas, questions, philosophies, recommendations, etc. Parents whose kids are attending regular schools but want to talk about learning materials are also welcome.

One request: Please no knee-jerk bashing of public schools or teachers. (I just find this really annoying.) Thoughtful, well-reasoned critique of mainstream schooling are fine, but let’s try to focus on the homeschooling.

This week’s successes:

DK Workbooks: Coding with Scratch (workbook) has been an amazing success.

Like many parents, I thought it’d be useful to learn some basic coding, but have no idea where to start. I once read HTML for dummies, but I don’t know my CSS from Perl, much less what’s best for kids.

After a bit of searching, I decided to try the the DK Coding with Scratch series. (This particular workbook is aimed at kids 6-9 yrs old, but there are others in the series.)

Scratch is a free, simple, child-friendly coding program available online at https://scratch.mit.edu/. You don’t need the workbook to use Scratch, (it’s just a helpful supplement.) There are also lots of helpful Youtube videos for the enterprising young coder.

Note: my kids really want to code because they want to make their own video games.

In general, I have found that toys and games that claim they will teach your kids to code actually won’t. (Eg, Robot Turtles.) Some of these games are a ton of fun anyway, I just wouldn’t expect to become a great coder that way.

Professor Astro Cat’s Frontiers of Space is as good as it looks. Target market is 8-11 years old. There’s a lot of information per page, so we’re reading and discussing a few pages each day.

There are two other books in the series, Professor Astro Cat’s Intergalactic Activity Book, which I’m hoping will make a good companion to this one, and Astro Cat’s Atomic Adventure, which looks like it fills the desperately needed “quantum physics for kids” niche.)

I’m still trying to figure out how to do hands-on science activities without spending a bundle. Most of the “little labs” type science kits look fun, but don’t pack a lot of educational bang for your buck. For example, today we built a compass (it cost $10 at the toy store, not the $205 someone is trying charge on Amazon.) This was fun and I really like the little model, but it also took about 5 minutes to snap the pieces together and we can’t actually carry it around to use it like a real compass.

Plus, most of these labs are basically single-use items. I like toys with a sciency-theme, but they’re too expensive to run the whole science curriculum off of.

Oh, sure, I hand them a page of math problems and they start squawking at me like chickens. But bedtime rolls around and they’re like, “Where’s our Bedtime Math? Can’t we do one more page? One more problem? Please?”

There are only three math problems every other page (though this does add up to over 100 problems,) the presentation is fun, and the kids like the book better than going to sleep.

The book offers easy, medium, and hard problems in each section, so it works for kids between the ages of about 4 and 10.

There’s an inherent tension in education between emphasizing subjects that kids are already good at and working on the ones they’re bad at. The former gives kids a chance to excel, build confidence, and of course actually get good at something, while the latter is often an annoying pain in the butt but nevertheless necessary.

 

Since we’ve just started and are still getting in the swing of things, I’m trying to focus primarily on the things they’re good at and enjoy and have just a little daily focus on the things they’re weak at.

I’d like to find a good typing tutor (I’ll probably be trying several out soon) because watching the kids hunt-and-peck at the keyboard makes my hair stand on end. I’d also like to find a good way to hold up workbooks next to the computer to make using the DK books easier.

That’s about it, so I’ll open the floor to you guys.

The big 6 part 6: The Vigesimal Olmecs

Olmec civilization heartland
Olmec civilization heartland

It appears that the Olmecs–our final civilization in this series (1500-400 BC)–had a vigesimal, or base 20, counting system.

Counting is one of those things that you learn to do so young and so thoroughly that you hardly give it a second thought; after a few hiccups around the age of five, when it seems logical that 11=2, the place value system also becomes second nature. So it is a bit disconcerting to realize that numbers do not actually divide naturally into groups of ten, that’s just a random culturally determined thing that we happen to do. (Well, it isn’t totally random–ten was probably chosen because our ancestors were counting on their fingers.)

Stela C, from Tres Zapotes, showing the date September 1, 32 BCE
Stela C, from Tres Zapotes, showing the date September 1, 32 BCE

But plenty of societies throughout history have used other bases. The Yuki of California used base 8 (they counted the spaces between fingers;) the Chumash use(d) base 4; Gumatj uses base 5. There are also reports of bases 12, 15, 25, 32, and 6. (And many hunter-gatherer societies never really developed words for numbers over three or so, though they easily employed phrases like “three threes” to mean “nine.”)

The Yoruba, Olmec, Maya, Aztec, Tlingit, Inuit, Bhutanese, Atong, Santali, Didei, Ainu all use (or used) base 20. Wikipedia suggests that the Mayans may have used their fingers and toes to count; I suggest they used the knuckles+fingertips on one hand, or in a sort of impromptu place-value system, used the fingers of one hand to represent 1-5, and the fingers of the other hand to represent completed groups of five. (eg, 3 fingers on your left hand = 3; 3 fingers on your right hand = 15.)

Mayan Numerals
Mayan Numerals

Everything I have seen of reliable genetics and anthropology suggests that the Olmecs and Mayans were related–for example, one of the first known Mayan calendars/Mayan dates was carved into the Mojarra Stela by the “Epi-Olmec” people who succeeded the Olmecs and lived in the Olmec city of Tres Zapotes. Of course this does not mean that the Olmecs themselves developed the calendar or written numbers, (though they could have,) but it strongly implies that they had the same base-20 counting system.

You can compare for yourself the numbers found on the Tres Zapotes stela (above) and the Mayan numerals (left.)

In base-10, we have special words for multiples of 10, like ten, twenty, ninety, hundred, thousand, etc. In a base-20 system, you have special words for multiples of 20, like twenty, (k‘áal, in Mayan;) forty, (ka’ k’áal, or “two twenties;”) four hundred, (bak😉 8,000, (pic😉 160,000 (calab;) etc.  Picture 13

Wikipedia helpfully provides a base-20 multiplication table, just in case you ever need to multiply in base-20.

The Olmecs, like the Egyptians and Sumerians, produced art (particularly sculptures,) monumental architecture, (pyramids,) and probably had writing and math. They raised corn, chocolate, (unsweetened,) squash, beans, avocados, sweet potatoes, cotton, turkeys, and dogs. (It appears the dogs were also eaten, “Despite the wide range of hunting and fishing available, midden surveys in San Lorenzo have found that the domesticated dog was the single most plentiful source of animal protein,[93]” possibly due to the relative lack of other domesticated animals, like cows.)

Cocoa pods
Cocoa pods

They also appear to have practiced ritual bloodletting (a kind of self-sacrifice in which the individual makes themselves bleed, in this case often by drawing sharp objects through their tongues, ears, or foreskins, or otherwise cutting or piercing these,) and played the Mesoamerican ballgame popular later with the Mayans and Aztecs. Whether these practices spread via cultural diffusion to other Meoamerican cultures or simply indicate some shared cultural ancestry, I don’t know.

Their sculptures are particularly interesting and display a sophisticated level of artistic skill, especially compared to, say, Norte Chico (though in its defense, Norte Chico did come earlier):

 

 

Another Olmec head
Another Olmec head
"The Wrestler," Olmec statue
“The Wrestler,” Olmec statue
Mosaic, La Venta
Mosaic, La Venta
Colossal Head of San Lorenzo,, Olmec
Colossal Head of San Lorenzo,, Olmec
Bird-shaped jug
Bird-shaped jug
Olmec baby statue
Olmec baby statue
Olmec mask
Olmec mask
Man holding a were-jaguar baby
Man holding a were-jaguar baby
Indigenous Mexican man and Olmec statue
Indigenous Mexican man and Olmec statue, from Johnson’s Mystery Solved: Olmec and Transoceanic Contact

Frank Johnson, in his post Mystery Solved: Olmecs and Transoceanic Contact

A lot of people think the Olmec stone heads look a lot like Africans (and I can see why,) but–as lots of people have pointed out–they also look a lot like the local Indians who live in the area today, and so far I haven’t run across any genetic studies that indicate African DNA (which is quite distinctive) in any Native American population (aside from the DNA we all share from our common, pre-out-of-Africa ancestors, 70,000-100,000 years ago.) (There is one tiny isolated tribe over in Baja CA, [Mexico,] quite far from where the Olmecs lived, who do have some interesting DNA stuff going on that could indicate contact with Africa or somewhere else, but it could also just indicate random genetic mutation in an extremely isolated, small population. At any rate, they are irrelevant to the Olmecs.)

Frank Johnson, in his post Mystery Solved: Olmecs and Transoceanic Contact, goes through the laundry list of questionable claims about the Olmecs and does a great job of laying out various proofs against them. While I would not totally rule out the possibility of trans-Atlantic (or trans-Pacific) contact between various groups, just because human history is long and full of mysteries, the most sensible explanation of the origins and cultural development of Olmec society is the simplest: the Olmecs were a local indigenous people, probably closely related to most if not all of their neighbors, who happened to start building cities and pyramids.

 

The Big 6 Civilizations (pt 2: Egypt)

1024px-Egypt.Giza.Sphinx.02

2. Egypt

I know I don’t have to tell you about Egyptian civilization, but did you know the Egyptians had maths?

Problem number 56 from the Rhind Mathematics Papyrus (dated to around 1650 BC):

Egyptian seked
Seked of the Great Pyramid

If you construct a pyramid with base side 12 [cubits] and with a seked of 5 palms 1 finger; what is its altitude?[1]

Most Egyptian geometry questions appear to deal with more mundane matters, like the dimensions of rectangular fields and round granaries, rather than pyramids. (The Egyptians had not yet worked out an exact formula for the area of a circle, but used octagons to approximate it.)

 

Picture 4A “pefsu” problem involves a measure of the strength of the beer made from a heqat of grain, called a pefsu.

pefsu = (the number of  loaves of bread [or jugs of beer]) / (number of heqats of grain used to make them.)

For example, problem number 8 from the Moscow Mathematical Papyrus (most likely written between 1803 BC and 1649 BC, but based on an earlier manuscript thought to have been written around 1850 BC):

Example of calculating 100 loaves of bread of pefsu 20:
If someone says to you: “You have 100 loaves of bread of pefsu 20 to be exchanged for beer of pefsu 4, like 1/2 1/4 malt-date beer,”
First calculate the grain required for the 100 loaves of the bread of pefsu 20. The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer. The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.
Calculate 1/2 of 5 heqat, the result will be 212. Take this 212 four times.
The result is 10. Then you say to him:
Behold! The beer quantity is found to be correct.[1]

“Behold! The beer quantity is found to be correct,” is one of the most amusing answers to a math problem I’ve seen.

Picture 5The Egyptians also used fractions and solved algebraic equations that we would write as linear equations, eg, 3/2 * x + 4 = 10.

But their multiplication and division was really weird, probably as a side effect of not yet having invented a place value system.

A. Let’s suppose you wished to multiply 9 * 19.

B. First we want to turn 9 into powers of 2.

C. The powers of 2 = 1, 2, 4, 8, 16, 32, 64, etc.

D. The closest of these to 9 is 8, and 9-8=1, so we turn 9 into 8 and 1.

E. Now we’re going to make a table using 1, 8, and 19 (from line A), like so:

1        19
2        ?
4         ?
8         ?

F. We fill in our table by doubling 19 each time:

1        19
2        38
4         76
8         152

E. Since we turned 9 into 1 and 8 (step D), we add together the numbers in our table that correspond to 1 and 8: 19 + 152 = 171.

Or to put it more simply, using more familiar methods:

9 * 19 = (1 +8) * 19 = (19 * 1) +(19 * 8) = (19 * 1) + (19 * 2 * 2 * 2) = 171

Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC), with number hieroglyphs
Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC), with number hieroglyphs

Now let’s do 247 * 250:

The closest power of 2 (without going over) is 128. 247 -128 = 119. 119 – 64 = 55. 55 – 32 = 23. 23 – 16 = 7. 7 – 4 = 3. 3 – 2 = 1. Whew! So we’re going to need 128, 64, 32, 16, 4, 2, and 1, and 250.

Let’s arrange our table, with the important numbers in bold (in this case, it’s :

1       250
2        ?
4         ?
8         ?
16       ?
32       ?
64       ?
128      ?

So, doubling 250 each time, we get:

1       250
2       500
4       1000
8        2000
16     4000
32     8000
64     16,000
128    32,000

Adding together the bold numbers in the second column gets us 61,750–and I probably don’t need to tell you that plugging 247 * 250 into your calculator (or doing it longhand) also gives you 61,750.

The advantage of this system is that the Egyptians only had to memorize their 2s table. The disadvantages are pretty obvious.

Berlin Papyrus
Berlin Papyrus

See also the Lahun Mathematical Papyri, the Egyptian Mathematical Leather Roll, the Akhmim wooden tablets, the Reisner Papyrus, and finally the Papyrus Anastasi I, which is believed to be a fictional, satirical tale for teaching scribes–basically, a funny textbook, and the Berlin Papyrus 6619:

The Berlin Papyrus contains two problems, the first stated as “the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other.”[6] The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x2 + y2 = 100 and x = (3/4)y reduce to the single equation in y: ((3/4)y)2 + y2 = 100, giving the solution y = 8 and x = 6.

Some quick notes on the big six civilizations (pt. 1)

Picture 4

ff23e2c73822050c646f06efd7503a4b

Proto-writing:

Chinese proto-writing
Chinese proto-writing

 

220px-Tartaria_amulet

European proto-writing
European proto-writing

 

Indus Valley seals
Indus Valley seals
Indus valley seal impression, possibly script
Indus valley seal impression, possibly script

 

 

 

The spread of agriculture
The spread of agriculture

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

wells2

1. Mesopotamia (Sumer):
fertile-crescent-ted-mitchellSumer (/ˈsmər/)[note 1] was the first ancient urban civilization in the historical region of southern Mesopotamia, modern-day southern Iraq, during the Chalcolithic and Early Bronze ages, and arguably the first civilization in the world.[1]

Proto-writing in the region dates back to c. 3500 BC. The earliest texts come from the cities of Uruk and Jemdet Nasr and date back to 3300 BC; early cuneiform writing emerged in 3000 BC.[2]

Cities of Sumer
Cities of Sumer

Modern historians have suggested that Sumer was first permanently settled between c. 5500 and 4000 BC by a West Asian people who spoke the Sumerian language (pointing to the names of cities, rivers, basic occupations, etc., as evidence), a language isolate.[3][4][5][6] …

Sumerian culture seems to have appeared as a fully formed civilization, with no pre-history. …

Uruk, one of Sumer’s largest cities, has been estimated to have had a population of 50,000-80,000 at its height;[28] given the other cities in Sumer, and the large agricultural population, a rough estimate for Sumer’s population might be 0.8 million to 1.5 million. The world population at this time has been estimated at about 27 million.[29]…

Babylonian math homework
Babylonian math homework*

The Sumerians developed a complex system of metrology c. 4000 BC. This advanced metrology resulted in the creation of arithmetic, geometry, and algebra. From c. 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[45] The period c. 2700 – 2300 BC saw the first appearance of the abacus, and a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[46] The Sumerians were the first to use a place value numeral system. … They were the first to find the area of a triangle and the volume of a cube.[47] …

* “Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which is good to about six decimal digits.
1 + 24/60 + 51/602 + 10/603 = 1.41421296… The tablet also gives an example where one side of the square is 30, and the resulting diagonal is 42 25 35 or 42.4263888…”

Continuing on:

Sumerian tablet recording the allocation of beer
Sumerian tablet recording the allocation of beer

Commercial credit and agricultural consumer loans were the main types of loans. The trade credit was usually extended by temples in order to finance trade expeditions and was nominated in silver. The interest rate was set at 1/60 a month (one shekel per mina) some time before 2000 BC and it remained at that level for about two thousand years.[49] Rural loans commonly arose as a result of unpaid obligations due to an institution (such as a temple), in this case the arrears were considered to be lent to the debtor.[50] They were denominated in barley or other crops and the interest rate was typically much higher than for commercial loans and could amount to 1/3 to 1/2 of the loan principal.[49]

Periodically “clean slate” decrees were signed by rulers which cancelled all the rural (but not commercial) debt and allowed bondservants to return to their homes. … The first known ones were made by Enmetena and Urukagina of Lagash in 2400-2350 BC. According to Hudson, the purpose of these decrees was to prevent debts mounting to a degree that they threatened fighting force which could happen if peasants lost the subsistence land or became bondservants due to the inability to repay the debt.[49] …

Examples of Sumerian technology include: the wheel, cuneiform script, arithmetic and geometry, irrigation systems, Sumerian boats, lunisolar calendar, bronze, leather, saws, chisels, hammers, braces, bits, nails, pins, rings, hoes, axes, knives, lancepoints, arrowheads, swords, glue, daggers, waterskins, bags, harnesses, armor, quivers, war chariots, scabbards, boots, sandals, harpoons and beer. The Sumerians had three main types of boats:

  • clinker-built sailboats stitched together with hair, featuring bitumen waterproofing
  • skin boats constructed from animal skins and reeds
  • wooden-oared ships, sometimes pulled upstream by people and animals walking along the nearby banks

… The Sumerians’ cuneiform script is the oldest (or second oldest after the Egyptian hieroglyphs) which has been deciphered (the status of even older inscriptions such as the Jiahu symbols and Tartaria tablets is controversial).

reconstructed Neo-Sumerian Great Ziggurat of Ur, near Nasiriyah, Iraq
Reconstructed Neo-Sumerian Great Ziggurat of Ur, near Nasiriyah, Iraq

Lamassu Designs has made a lovely infographic on the Sumerian/Mesopotamian calendar/numerical system, which for some reason is failing to download properly. So I’m screencapping it for you:

by Lamassu Design, part 1

by Lamassu Design, part 2

Lamassu Design, part 3

Lamassu Design, part 4

Lamassu Design, part 5

Lamassu Design, part 6

by Lamassu Design, part 7

by Lamassu Design, part 8

by Lamassu Design, part 9

Lamassu Design, part 10

Lamassu Design, part 11

Lamassu Design, part 12

Lamassu Design, part 13

Lamassu Design, part 14

Lamassu Design, part 15

Lamassu Design, part 16 Lamassu Design, part 17

 

1280px-Ur_mosaic  Standard_of_Ur_chariots

marsh near the mouths of the Tigris and Euphrates rivers
marsh near the mouths of the Tigris and Euphrates rivers
Reconstructed Sumerian finery
Reconstructed Sumerian finery

 

 

 

 

 

 

 

 

 

 

 

Do small families lead to higher IQ?

Okay, so this is just me thinking (and mathing) out loud. Suppose we have two different groups (A and B) of 100 people each (arbitrary number chosen for ease of dividing.) In Group A, people are lumped into 5 large “clans” of 20 people each. In Group B, people are lumped in 20 small clans of 5 people each.

Each society has an average IQ of 100–ten people with 80IQs, ten people with 120IQs, and eighty people with 100IQs. I assume that there is slight but not absolute assortative mating, so that most high-IQ and low-IQ people end up marrying someone average.

IQ pairings:

100/100    100/80    100/120    80/80    120/120 (IQ)

30                 9                9                 1               1            (couples)

Okay, so there should be thirty couples where both partners have 100IQs, nine 100/80IQ couples, nine 100/120IQ couples, one 80/80IQ couple, and one 120/120IQ couple.

If each couple has 2 kids, distributed thusly:

100/100=> 10% 80, 10% 120, and 80% 100

120/120=> 100% 120

80/80 => 100% 80

120/100=> 100% 110

80/100 => 100% 90

Then we’ll end up with eight 80IQ kids, eighteen 90IQ, forty-eight 100IQ, eighteen 110 IQ, and 8 120IQ.

So, under pretty much perfect and totally arbitrary conditions that probably only vaguely approximate how genetics actually works (also, we are ignoring the influence of random chance on the grounds that it is random and therefore evens out over the long-term,) our population approaches a normal bell-curved IQ distribution.

Third gen:

80/80  80/90  80/100  90/90  90/100  90/110  100/100  100/110  100/120  110/110  110/120  120/120

1             2            5             4            9             2              6                9               5              4             2             1

2 80         4 85      10 90      8 90     18 95      4 100       1,4,1       18 105     10 110        8 110       4 115        2 120

3 80, 4 85, 18 90, 18 95, 8 100, 18 105, 18 110, 4 115, and 3 120. For simplicity’s sake:

7 80IQ, 18 90IQ, 44 100IQ, 18 110IQ, and 7 120IQ.

Not bad for a very, very rough model that is trying to keep the math very simple so I can write it blog post window instead of paper, though clearly 6 children have gotten lost somewhere. (rounding error???)

Anyway, now let’s assume that we don’t have a 2-child policy in place, but that being smart (or dumb) does something to your reproductive chances.

In the simplest model, people with 80IQs have zero children, 90s have one child, 100s have 2 children, 110s have 3 children, and 120s have 4 children.

oh god but the couples are crossed so do I take the average or the top IQ? I guess I’ll take average.

Gen 2:

100/100    100/80    100/120    80/80    120/120 (IQ)

30                 9                9                 1               1            (couples)

60 kids        9 kids       27 kids       0              4 kids

6, 48, 6

So our new distribution is six 80IQ, nine 90IQ, forty-eight 100IQ, twenty-seven 110IQ, and ten 120IQ.

(checks math oh good it adds up to 100.)

We’re not going to run gen three, as obviously the trend will continue.

Let’s go back to our original clans. Society A has 5 clans of 20 people each; Society B has 20 clans of 5 people each.

With 10 high-IQ and 10 low-IQ people per society, each clan in A is likely to have 2 smart and 2 dumb people. Each clan in B, by contrast, is likely to have only 1 smart or 1 dumb person. For our model, each clan will be the reproductive unit rather than each couple, and we’ll take the average IQ of each clan.

Society A: 5 clans with average of 100 IQ => social stasis.

Society B: 20 clans, 10 with average of 96, 10 with average of 106. Not a big difference, but if the 106s have even just a few more children over the generations than the 96s, they will gradually increase as a % of the population.

Of course, over the generations, a few of our 5-person clans will get two smart people (average IQ 108), a dumb and a smart (average 100), and two dumb (92.) The 108 clans will do very well for themselves, and the 92 clans will do very badly.

Speculative conclusions:

If society functions so that smart people have more offspring than dumb people (definitely not a given in the real world,) then: In society A, everyone benefits from the smart people, whose brains uplift their entire extended families (large clans.) This helps everyone, especially the least capable, who otherwise could not have provided for themselves. However, the average IQ in society A doesn’t move much, because you are likely to have equal numbers of dumb and smart people in each family, balancing each other out. In Society B, the smart people are still helping their families, but since their families are smaller, random chance dictates that they are less likely to have a dumb person in their families. The families with the misfortune to have a dumb member suffer and have fewer children as a result; the families with the good fortune to have a smart member benefit and have more children as a result. Society B has more suffering, but also evolves to have a higher average IQ. Society A has less suffering, but its IQ does not change. Obviously this a thought experiment and should not be taken as proof of anything about real world genetics. But my suspicion is that this is basically the mechanism behind the evolution of high-IQ in areas with long histories of nuclear, atomized families, and the mechanism suppressing IQ in areas with strongly tribal norms. (See HBD Chick for everything family structure related.)

 

 

Is Capitalism the only reason to care about Intelligence?

Trophonius--ὃν οἱ θεοὶ φιλοῦσιν, ἀποθνῄσκει νέος.
Trophonius–ὃν οἱ θεοὶ φιλοῦσιν, ἀποθνῄσκει νέος.

Before we get started, I want to pause in memory of Henry Harpending, co-author (with Greg Cochran) of The 10,000 Year Explosion: How Civilization Accelerated Human Evolution and the blog West Hunter.

ὃν οἱ θεοὶ φιλοῦσιν, ἀποθνῄσκει νέος — he whom the gods love dies young. (Meander)

Harpending wasn’t particularly young, nor was his death unexpected, but I am still sad; I have enjoyed his work for years, and there will be no more. Steve Sailer has a nice eulogy.

In less tragic HBD-osphere news, it looks like Peter Frost has stopped writing his blog, Evo and Proud, due to Canadian laws prohibiting free speech. (There has been much discussion of this on the Frost’s posts that were carried over on Unz; ultimately, the antisemitism of many Unz commentators made it too dangerous for Frost to continue blogging, even though his posts actually had nothing to do with Judaism.)

Back to our subject: This is an attempt to answer–coherently–a friend’s inquiry.

  1. Why are people snobs about intelligence?
  2. Is math ability better than verbal?
  3. Do people only care about intelligence in the context of making money?

We’re going to tackle the easiest question first, #2. No, math ability is not actually better than verbal ability.

Imagine two people. Person A–we’ll call her Alice–has exceptional verbal ability. She probably has a job as a journalist, novelist, poet, or screenwriter. She understands other people’s emotions and excels at interacting verbally with others. But she sucks at math. Not just suck; she struggles counting to ten.

Alice is going to have a rough time handling money. In fact, Alice will probably be completely dependent on the other people around them to handle money for them. Otherwise, however, Alice will probably have a pretty pleasant life.

Of course, if Alice happened to live in a hunter-gatherer society where people don’t use numbers over 5, she would not stand out at all. Alice could be a highly respected oral poet or storyteller–perhaps her society’s version of an encyclopedia, considered wise and knowledgeable about a whole range of things.

Now consider Person B–we’ll call her Betty. Betty has exceptional math ability, but can only say a handful of words and cannot intuit other people’s emotions.

Betty is screwed.

Here’s the twist: #2 is a trick question.

Verbal and mathematical ability are strongly correlated in pretty much everyone who hasn’t had brain damage (so long as you are looking at people from the same society). Yes, people like to talk about “multiple intelligences,” like “kinesthetic” and “musical” intelligence. It turns out that most of these are correlated. (The one exception may be kinesthetic, about which I have heard conflicting reports. I swear I read a study somewhere which found that sports players are smarter than sports watchers, but all I’m finding now are reports that athletes are pretty dumb.)

Yes, many–perhaps most–people are better at one skill than another. This effect is generally small–we’re talking about people who get A+ in English and only B+s in math, not people who get A+ in English but Fs in math.

The effect may be more pronounced for people at the extremes of high-IQ–that is, someone who is three standard deviations above the norm in math may be only slightly above average in verbal, and vice versa–but professional authors are not generally innumerate, nor are mathematicians and scientists unable to read and write. (In fact, their professions require constantly writing papers for publication and reading the papers published by their colleagues.)

All forms of “intelligence” probably rely, at a basic level, on bodily well-being. Your brain is a physical object inside your body, and if you do not have the material bits necessary for well-being, your brain will suffer. When you haven’t slept in a long time, your ability to think goes down the tubes. If you haven’t eaten in several days (or perhaps just this morning), you will find it difficult to think. If you are sick or in pain, again, you will have trouble thinking.

Healthy people have an easier time thinking, and this applies across the board to all forms of thought–mathematical, verbal, emotional, kinesthetic, musical, etc.

“Health” here doesn’t  just include things we normally associate with it, like eating enough vegetables and swearing to the dentist that this time, you’re really going to floss. It probably also includes minute genetic variations in how efficient your body is at building and repairing tissues; chemicals or viruses you were exposed to in-utero; epigenetics, etc.

So where does this notion that math and science are better than English and feelings come from, anyway?

A. Math (and science) are disciplines with (fairly) objective answers. If I ask you, “What’s 2+2?” we can determine pretty easily whether you got it correct. This makes mathematical ability difficult to fudge and easy to verify.

Verbal disciplines, by contrast, are notoriously fuzzy:

  riverrun, past Eve and Adam’s, from swerve of shore to bend 1
of bay, brings us by a commodius vicus of recirculation back to 2
Howth Castle and Environs. 3
    Sir Tristram, violer d’amores, fr’over the short sea, had passen- 4
core rearrived from North Armorica on this side the scraggy 5
isthmus of Europe Minor to wielderfight his penisolate war: nor 6
had topsawyer’s rocks by the stream Oconee exaggerated themselse 7
to Laurens County’s gorgios while they went doublin their mumper 8
all the time: nor avoice from afire bellowsed mishe mishe to 9
tauftauf thuartpeatrick: not yet, though venissoon after, had a 10
kidscad buttended a bland old isaac: not yet, though all’s fair in 11
vanessy, were sosie sesthers wroth with twone nathandjoe. Rot a 12
peck of pa’s malt had Jhem or Shen brewed by arclight and rory 13
end to the regginbrow was to be seen ringsome on the aquaface. 14
    The fall (bababadalgharaghtakamminarronnkonnbronntonner- 15
ronntuonnthunntrovarrhounawnskawntoohoohoordenenthur- 16
nuk!)

So. A+ or F-?

Or how about:

I scowl with frustration at myself in the mirror. Damn my hair – it just won’t behave, and damn Katherine Kavanagh for being ill and subjecting me to this ordeal. I should be studying for my final exams, which are next week, yet here I am trying to brush my hair into submission. I must not sleep with it wet. I must not sleep with it wet. Reciting this mantra several times, I attempt, once more, to bring it under control with the brush. I roll my eyes in exasperation and gaze at the pale, brown-haired girl with blue eyes too big for her face staring back at me, and give up. My only option is to restrain my wayward hair in a ponytail and hope that I look semi presentable.

Best-seller, or Mary Sue dreck?

And what does this mean:

Within that conflictual economy of colonial discourse which Edward Said describes as the tension between the synchronic panoptical vision of domination – the demand for identity, stasis – and the counterpressure of the diachrony of history – change, difference – mimicry represents an ironic compromise. If I may adapt Samuel Weber’s formulation of the marginalizing vision of castration, then colonial mimicry is the desire for a reformed, recognizable Other, as a subject of a difference that is almost the same, but not quite. Which is to say, that the discourse of mimicry is constructed around an ambivalence; in order to be effective, mimicry must continually produce its slippage, its excess, its difference. (source)

If we’re going to argue about who’s smartest, it’s much easier if we can assign a number to everyone and declare that the person with the biggest number wins. The SAT makes a valiant effort at quantifying verbal knowledge like the number of words you can accurately use, but it is very hard to articulate what makes a text so great that Harvard University would hire the guy who wrote it.

B. The products of science have immediately obvious, useful applications, while the products of verbal abilities appear more superficial and superfluous.

Where would we be today without the polio vaccine, internal combustion engines, or the transistor? What language would we be writing in if no one had cracked the Enigma code, or if the Nazis had not made Albert Einstein a persona non grata? How many of us used computers, TVs, or microwaves? And let’s not forget all of the science that has gone into breeding and raising massively more caloric strains of wheat, corn, chicken, beef, etc., to assuage the world’s hunger.

We now live in a country where too much food is our greatest health problem!

If I had to pick between the polio vaccine and War and Peace, I’d pick the vaccine, even if every minute spent with Tolstoy is a minute of happiness. (Except when *spoilers spoilers* and then I cry.)

But literature is not the only product of verbal ability; we wouldn’t be able to tell other people about our scientific discoveries if it weren’t for language.

Highly verbal people are good at communication and so help keep the gears of modern society turning, which is probably why La Griffe du Lion found that national per capita GDP correlated more closely with verbal IQ scores than composite or mathematical scores.

Of course, as noted, these scores are highly correlated–so the whole business is really kind of moot.

So where does this notion come from?

In reality, high-verbal people tend to be more respected and better paid than high-math people. No, not novelists–novelists get paid crap. But average pay for lawyers–high verbal–is much better than average pay for mathematicians. Scientists are poorly paid compared to other folks with similar IQs and do badly on the dating market; normal people frequently bond over their lack of math ability.

“Math is hard. Let’s go shopping!” — Barbie

Even at the elementary level, math and science are given short shrift. How many schools have a “library” for math and science exploration in the same way they have a “library” for books? I have seen the lower elementary curriculum; kindergarteners are expected to read small books and write full sentences, but by the end of the year, they are only expected to count to 20 and add/subtract numbers up to 5. (eg, 1+4, 2+3, 3-2, etc.)

The claim that math/science abilities are more important than verbal abilities probably stems primarily from high-math/science people who recognize their fields’ contributions to so many important parts of modern life and are annoyed (or angry) about the lack of recognition they receive.

To be Continued.

 

Why do Rh- People Exist?

Having the Rh- bloodtype makes reproduction difficult, because Rh- mothers paired with Rh+ fathers end up with a lot of miscarriages.*

The simplified version: Rh+ people have a specific antigen in their blood. Rh- people don’t have this antigen.

If a little bit of Rh+ blood gets into an Rh- person’s bloodstream, their immune system notices this new antibody they’ve never seen before and the immune response kicks into gear.

If a little bit of Rh- blood gets into an Rh+ person’s bloodstream, their immune system notices nothing because there’s nothing to notice.

During pregnancy, it is fairly normal for a small amount of the fetus’s blood to cross out of the placenta and get into the mother’s bloodstream. One of the effects of this is that years later, you can find little bits of their children’s DNA still hanging around in women’s bodies.

If the mother and father are both Rh- or Rh+, there’s no problem, and the mother’s body takes no note of the fetuses blood. Same for an Rh+ mother with an Rh- father. But when an Rh- mother and Rh+ father mate, the result is bloodtype incompatibility: the mother begins making antibodies that attack her own child’s blood.

The first fetus generally comes out fine, but a second Rh+ fetus is likely to miscarry. As a result, Female Rh- with Male Rh+ pairings tend not to have a lot of children. This seems really disadvantageous, so I’ve been trying to work out if Rh- bloodtype ought to disappear out over time.

Starting with a few simplifying assumptions and doing some quick back of the envelope calculations:

  1. We’re in an optimal environment where everyone has 10 children unless Rh incompatibility gets in the way.
  2. Blood type is inherited via a simple Mendelian model. People who are ++, +-, and -+ all have Rh+ blood. People with — are Rh-.
  3. We start with a population that is 25% ++, +-, -+, and –, respectively.

So our 1st generation pairings are:

F++/M++   F++/M+-   F++/M-+   F++/M–

F+-/M++    F+-/M+-    F+-/M-+    F+-/M–

F-+/M++    F-+/M+-    F-+/M-+    F-+/M–

F–/M++     F–/M+-      F–/M-+     F–/M–

Which gives us:

10++,           5++, 5+-       5+-, 5++     10+-

5++, 5-+      2.5++, 2.5+-, 2.5-+, 2.5–   2.5+-, 2.5++, 2.5–, 2.5 -+      5+-, 5–

5-+, 5++      2.5-+, 2.5–, 2.5++, 2.5+-    2.5–, 2.5-+, 2.5+-, 2.5++      5–, 5+-

1-+,         It’s complicated   It’s complicated   10–

or

50++,   40+-,   21-+,   30–,   and some quantity of “It’s complicated.”

For the F–/M+- pairings, any — children will live and most -+ children will die. Since we’re assuming 10 children, we’re going to calculate the odds for ten kids. Dead kids in bold; live kids plain.

Kid 1: 50% -+,                     50% —

Kid 2: 25% -+, 25% —       25% -+, 25% —

Kid 3: 25% -+, 25% —       12.5% -+, 12.5% —    12.5% -+, 12.5% —

Kid 4: 25% -+, 25% —       12.5% -+, 12.5% —     6.3% -+, 6.3% —      6.3% -+, 6.3% –

Kid 5: 25% -+, 25% —        12.5% -+, 12.5% —    6.3% -+, 6.3% —       3.1% -+, 3.3% —    3.1% -+, 3.1% —

Obvious pattern is obvious: F–/M+- pairings lose 25% of their second kids, 37.5% of their third kids, 43.3% of their fourth kids, 46.4% of their fifth kids, etc, on to about 50% of their 10th kids.

Which I believe works out to an average of 5–, 1+-

The outcomes for F–/M-+ pairings are the same, of course: 5–, 1+-

So this gives us a total of:

50++, 41+-, 22-+, 40–,  or  33% ++, 27% +-, 14% -+, 26% —  (or, 54% of the alleles are + and 46% are -).

(This assumes, of course, that people cannot increase their number of pregnancies.)

Running the numbers through again (I will spare you my arithmetic), we get:

35% ++, 32% +-, 11.8%-+, 21.4% —  (or, 57% of alleles are + and 43% are – ).

I’m going to be lazy and say that if this keeps up, it looks like the –s should become fewer and fewer over time.

But I’ve made a lot of simplifying assumptions to get here that might be affecting my outcome. For example, if people only have one kid, there’s no effect at all, because only second children on down get hit by the antibodies. Also, people can have additional pregnancies to make up for miscarriages. 20 pregnancies is obviously pushing the limits of what humans can actually get done, but let’s run with it.

So in the first generation, F–/M+- => 9–, 1+-  ; F–/M-+ => 9–, 1-+ (that is, the extra pregnancies result in 8 extra — children.) The F–/M++ pairing still results in only one -+ child.

This gives us 50++, 41+-, 22-+, 48– children, or 31%++, 25%+-, 13.7%, 30%– (or 51% + vs 49% – alleles.)

At this point, the effect is tiny. However, as I noted before, having 20 pregnancies is a bit of a stretch for most people; I suspect the effect would still be generally felt under normal conditions. For example, I know an older couple who suffered Rh incompatibility; they wanted 4 children, but after many miscarriages, only had 3.

Which leads to the question of why Rh-s exist at all, which we’ll discuss tomorrow.

 

*Lest I worry anyone, take heart: modern medicine has a method to prevent the miscarriage of Rh+ fetuses of Rh- mothers. Unfortunately, it requires an injection of human blood serum, which I obviously find icky.

 

 

 

 

Bi-modal brains?

But... the second equation makes perfect sense.
But… the second equation makes sense.

So I have this co-woker–we’ll call her Delta. (Certain details have been changed to protect the privacy of the innocent.) Delta is an obviously competent, skilled worker who has succeeded at her job in a somewhat technical field for many years. She has multiple non-humanities degrees or accredidations. And yet, she frequently says things that are mind-numbingly dumb and make me want to bang my head on my desk.

To be fair, everybody makes mistakes and says incorrect things sometimes; maybe she thinks the exact same thing about me. Also, I have no real perspective on how dumb people think, because I haven’t spent much of my life talking to them. Even the formerly homeless people I know can carry on a layman’s discussion of quantum physics.

At any rate, I don’t actually think Delta is dumb. Instead, I think she has, essentially, two brain modes: Feeling Mode and Logic Mode.

Feeling Mode happens to be her default; she can do Logic Mode perfectly well, but she has to concentrate to activate it. If Logic Mode isn’t on, then things just get automatically processed through Feelings Mode and, as a result, don’t always make sense.

When Logic Mode is on, she does quite fine–her career, after all, is dependent on her rational, logical abilities, above-average math skills, etc. But her job is just that, not a passion, not something she’d do if it didn’t put food on the table. When she is in default mode, her brain just doesn’t make logical connections, notice patterns (especially meta-patterns), or otherwise understand a lot of the stuff going on around her. And her inability to judge distances/estimate sizes just makes me cringe.

My conversation topics typically go over like lead balloons.

In a recent Stanford Magazine article, Content to Code? in which Marissa Messina discusses her decision to major in computer science:

BEFORE STANFORD, I’d never heard the term “CS.” When my pre-Orientation mates used it repeatedly during our technology-free week of hiking in Yosemite prior to the start of freshman year, I had to ask them what it stood for. But their matter-of-fact response—”computer science”—was still a foreign concept to me. …

“Nonetheless, I celebrate my decision to develop my technical side. Although it does not come naturally to me, in Bay Area culture, knowing how to code feels like a prerequisite to existing. …

“I quickly learned through get-to-know-you conversations that being a “techie” was inherently cooler than being a “fuzzie,” and that social standard plus rumors of superior job prospects for engineers began to make me question my plan to major in psychology.

“Three years later, here I am, close to graduating and capable of coding. Now what?

“I certainly don’t imagine myself thriving as a professional programmer, because thinking in syntactically flawless computer-speak remains a wearisome process for me. … “

How on Earth does anyone arrive at Stanford without knowing that computer science exists?

Messina illustrates my theory rather well. She can go into logic mode, she can write code well enough to major in CS at Stanford, but it does not come naturally to her and she finds it rather unpleasant. She is only doing it because, back in freshman year, someone said her job prospects would be better with a CS degree. Now she realizes that she doesn’t actually want to do CS for a full-time job.

I suspect that most people operate primarily in Feelings Mode, and may be even worse than my co-worker at activating Logic Mode. Some may not have an operative Logic Mode at all; a few people may not have a Feeling Mode, but that seems less common. Feelings are instinctual, irrational, and messy. They exist because they are useful, but that does not mean they make logical sense.

For example, let’s suppose an out-of-control train is racing toward a group of schoolchildren who’ve been tied to the railroad tracks, but if you push a 9-foot tall man in heavy plate mail in front of the train, his death will save the children.

People operating in Logic Mode start debating the virtues of Kant’s Categorical Imperative verses Mill’s Utilitarianism.

People operating in Feelings Mode want to know what kind of psycho came up with a fucked up question like that. Children tied to the train tracks? Murdering an innocent bystander by pushing him in front of the train? Why are you fuckers debating this? Are you all sick in the head?

When Feeling people switch over into Logic Mode, I suspect it exerts some cost on them: that is, they can do it, but they don’t really like it. It’s uncomfortable, unpleasant, and sometimes exhausting. So most of the time, they prefer to be in default mode.

So there are things that they can understand in Logic Mode, but since they find the whole business unpleasant, they prefer to ignore such conclusions if they possibly can. This probably makes it very difficult to get people to make any kind of decisions involving unpleasant scenarios + data. The unpleasantness itself of the scenario breaks them out of Logic Mode and into Feeling Mode, and then the whole business is flushed down the toilet because someone goes into a screaming fit because you hurt their feelings with your data.

Earlier this morning, I happened across this “Systematizing Quotient” Quiz that HBD Chick linked to. Obviously the quiz has certain drawbacks, like user bias and the difficulty of comparing oneself to others (do I know more or less about car engines than other people? I probably know less about them than most men, but since I can diagram how an engine works and explain it, do I know more than the average woman? Where do I fall on a population scale? And what if I wouldn’t research something before buying it because I already know all about it, or because I think the brands available on the market are similar enough that the time spent resourcing would not be cost-effective?) but I thought I’d try it, anyway.

I scored in the 61-80 range, which is not terribly surprising. What’s weird is just how low everyone else scores, since the averages are 24 and 30 for women and men, respectively, and it’s not like the scale goes down to -50 or anything.

At any rate, when Delta started talking about how much she hates the Common Core math, well, I was curious. I did some digging and came up with problems like the one at the top of the screen, generally accompanied by a bunch of comments from parents like, “What are they even doing?” and “I have no idea what that is!” and “That makes no sense!” And I just look at them all like, Wow, you can’t figure out that 5+2+10+10+10=37?

Sure, math is a recently evolved trait and all, but those sorts of comments still vaguely surprise me.

IQ probably intersects the two modes via a separate axis. That is, a high-IQ Feelings Person might be able to concentrate enough of their mental resources to out-math a low-IQ Logic person, and vice versa, a high-IQ Logic Person might be able to concentrate enough mental resources to out-feel a Feeling Person. (For example, by reading a book about what various facial expressions mean and then using that knowledge in real life.) Delta, for example, could probably figure out the problem after a while, but would still say it’s a terrible problem.

There was a conversation around here somewhere about a recent paper that came out claiming that the discrepancy between the number of men and women in high-end mathematics was due to not enough girls taking rigorous math courses in middle school. Well, I don’t know about the middle schools where the paper was published, but my middle school only had one math class, and we all took it, so I don’t think that’s exactly the problem. More likely, cognitive differences just happen to be manifesting themselves in Middle School, and the math geniuses are starting to outshine people who are smart and hard working but not geniuses.

In the conversation, someone remarked that while women (or in this case, girls,) they’ve known can do math perfectly well, they tend not to enjoy it, and prefer doing other things, whereas the men they know are more or less forced to do it because their brains just happen to automatically look for patterns. This was the original inspiration for this post; the idea that someone might be able to switch back and forth between two modes, but would generally prefer one, while someone else might generally prefer the other. I might call it “Logic Mode” and The Guardian might call it “Systematizing Mode”, but they’re both basically the same.

If this is true, most people may not operate in Feeling Mode, but most women do. On the other hand, it may be that only a small sub-set of men operate primarily in Logic Mode, either, but they happen to be a larger sub-set than the sub-set of women who operate primarily in Logic Mode. Since I don’t talk to most people (no one possibly could,) and my real-life conversations are largely limited to other women, I am curious about your personal observations.